The subject of this paper is the dynamics of wave motion in
the two-dimensional Kelvin-Helmholtz problem for an interface between
two immiscible fluids of different densities. The difference of the mean
flow between the two fluid bodies is taken to be zero, and the effects of
surface tension are neglected. We transform the problem to Birkhoff normal
form, in which a precise analysis can be made of classes of resonant
solutions. This paper studies standing-wave solutions of the fourth-order
normal form in particular detail. We find that that there are families of
invariant resonant subsystems, which are nevertheless integrable. Within
these families we describe the periodic and the time quasi-periodic standing
waves, and determine their stability or instability. In particular we
show that for a certain range of densities, a basic time-periodic standing
wave with principal wave number k is unstable to modes with principal
wave numbers k=4 and 9k=4, and we calculate the Lyapunov exponent of
the instability. We furthermore show that the stable and unstable manifolds
to these periodic solutions of the Birkhoff normal form are connected
by a homoclinic orbit. This instability mechanism, as well as others that
we describe, appears to be new, and its description is possible because of
the precision afforded by the normal form. These results contrast with the
case of the water wave problem described by Dyachenko & Zakharov [1]
and Craig & Worfolk [2], where the fourth-order Birkhoff normal form
is an integrable system, with all orbits undergoing stable almost-periodic
motion, and instabilities arise only in normal forms to higher order.